## Abstract

For a matroid M of rank r on n elements, let b(M) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show that Ω(1/n)≤1−b(M)≤O(log(n)3/n)asn→∞ for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U
_{k,2k}-minor, whenever k≤O(log(n)), (2) have girth ≥Ω(log(n)), (3) have Tutte connectivity ≥Ω(log(n)), and (4) do not arise as the truncation of another matroid. Our argument is based on a refined method for writing compressed descriptions of any given matroid, which allows bounding the number of matroids in a class relative to the number of sparse paving matroids.

Original language | English |
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Pages (from-to) | 955–985 |

Number of pages | 31 |

Journal | Combinatorica |

Volume | 38 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Aug 2018 |